Using Algebraic Geometry for Multivariate Polynomial Interpolation

Abstract

Interpolation provides a direct way of fitting analytic functions to sampled data. Chapter 8 is motivated by computational efficiency and deals with polynomials rather than arbitrary analytic forms. We distinguish between multivariate polynomial functions

$$F:x_n = f_1 \left( {x_1 , \ldots ,x_{n - 1} } \right)$$

multivariate rational functions

$$\Re :x_n = \frac{{f_1 \left( {x_1 , \ldots ,X_{n - 1} } \right)}} {{f_2 \left( {x_1 , \ldots ,X_{n - 1} } \right)}}$$

and polynomial algebraic functions or implicitly defined hypersurfaces

$$H:f_1 \left( {x_1 , \ldots ,X_n } \right) = 0$$

where all f _i are multivariate polynomials with coefficients in ℝ. While prior work on interpolation has dealt with multivariate polynomial functions F and rational functions ℬ, see for example References 1–5, little work has been reported on interpolation with implicitly defined hypersur-faces ℋ. See References 6 and 15 which summarizes prior work on implicit surface interpolation in three dimensions and provides several additional references.